Updates to Differential Equation (de) Solvers In

8/14/2019 Updates to Differential Equation (de) Solvers In

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Updates to Differential Equation (DE) Solvers in Maple 12


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Summary


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The differential equation theme for Maple 12 (exact solutions) is the exploration of different

transformations for mapping given equations into ones the system knows how to solve. Using this

approach, a new algorithm was developed by our research team, for 3rd order linear Ordinary

Differential Equations (ODEs), that solves for the first time the entire 3F2, 2F2, 1F2 and 0F2hypergeometric classes of equations in terms of hypergeometric and MeijerG functions [1]. The

same idea is used for solving an entire family of 2nd ordernonlinearODEs parametrized by two

arbitrary functions, where the solutions can be derived from the solutions of a related 3rd orderlinearODE. A similar approach is used to solve the entire nonlinear 1st orderAbel AIL class

exploring hypergeometric functions.


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Two new commands, rational_equivalent and ODEInvariants, have been added to the DEtools

package, the latter returning the so-called Wilcynski Invariants forlinearODEs as well as an

innovative derivation based on them that results in invariants fornonlinearODEs.


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Regarding numerical solutions for ODEs, there are a significant number of improvements such as the

new ability to handle user-defined events, parametrized problems, and the definition of discrete

variables.


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Exact Solutions


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Ordinary Differential Equations (ODEs)


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New solutions in terms of hypergeometric functions for 3rd order linear equations


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New solving algorithms for the entire 3F2, 2F2, 1F2 and 0F2 hypergeometric classes of equations were

implemented. These new solvable classes are the ones you obtain from the standard pFq equations,


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> PDEtools:-declare(y(x), prime = x);


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( )y x will now be displayed as y


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derivatives with respect to x of functions of one variable will now be displayed with '


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> for pFq in [hypergeom([], [d,e], x), hypergeom([c], [d,e], x),hypergeom([b,c], [d,e], x), hypergeom([a,b,c], [d,e], x)]


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do


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FunctionAdvisor(DE, pFq, y(x));


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od;


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,=y ( )hypergeom , ,[ ] [ ],d e x

=y''' + +

e d y'

x2( ) e 1 d y''

x

y

x2


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,=y ( )hypergeom , ,[ ]c [ ],d e x

=y''' + +

( ) x e d y' x2

( ) e 1 d y'' x

c y

x2


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=y ( )hypergeom , ,[ ],b c [ ],d e x ,

=y''' + +

( )( )+ +1 b c x e d y' x2

( ) x e 1 d y'' x

c b y

x2


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=y ( )hypergeom , ,[ ], ,a b c [ ],d e x y''' =,

( )+( )( ) 1 b c a ( )+c 1 ( )+b 1 x e d y' x2 ( ) +1 x

( )+ + +( ) 3 c b a x e 1 d y'' x ( ) +1 x

+

c b a y

x2 ( ) +1 x


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by changing variables using a composition of the general transformations


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> x -> R(x), y -> P(x)*y;


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,x ( )Rx y ( )P x y


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where R(x) is a rational function and P(x) is arbitrary, and


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> x -> (a*x^k+b)/(c*x^k+d);


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x+a xk b+c xk d


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that is, a power composed with a Mobius transformation where a, b, c, d, kare arbitrary constants with

respect tox. Hence, the Maple system can now solve the equivalence problem to these hypergeometric

equations under rational transformations and also in the case where these transformations are composed

with fractional or symbolic powers. These results by our research team are new in the literature [1].


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Examples


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The following equation admits three pFq hypergeometric solutions computed after determining a

rational transformation relating the equation to the 0F2 equation (the first one in the output of the

FunctionAdvisorabove)


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> diff(y(x), x, x, x) = - 6/x*(x-3)/(x-2)*diff(y(x), x, x) -6/x^2*(x^2-6*x+10)/(x-2)^2*diff(y(x), x) - (x-1)*(x-2)^3/x^11*y(x);


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=y''' 6 ( )x 3 y''x ( )x 2

6 ( ) +x2 6x 10 y'x2 ( )x 2 2

( ) +1 x ( )x 2 3yx11


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> dsolve(%);


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y _C1

hypergeom , ,[ ]

,

1

2

3

4

( ) +1 x 4

64x8=

_C2 ( ) +1 x

hypergeom , ,[ ]

,

3

4

5

4

( ) +1 x 4

64 x8

x2+

_C3 ( ) +1 x

hypergeom , ,[ ]

,

5

4

3

2

( ) +1 x 4

64x8 +1 x

x4+


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When the hypergeometric parameters are such that there are not three independent hypergeometric

solutions, the new algorithm uses MeijerG functions to represent the missing ones, as in:


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> diff(y(x), x, x, x) = -(12*x^2-5*x+1)/x^2/(-1+2*x)*diff(y(x), x, x)+ 1/x^3*(20*x^2-c-5*x+2*c*x-24*x^3)/(-1+2*x)^2*diff(y(x), x) +c*y(x)/x^4/(-1+2*x)^2;


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=y''' + +( ) +12x2 5x 1 y''

x2 ( ) +1 2x( ) + 20x2 c 5x 2 c x 24x3 y'

x3 ( ) +1 2x 2c y

x4 ( ) +1 2x 2


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> dsolve(%);


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y _C1

hypergeom , ,[ ]c [ ]1

+1 2xx

=

_C2 MeijerG , ,[ ],[ ] +c 1 [ ] [ ],[ ],0 0 [ ]

+1 2xx

+

_C3

MeijerG , ,[ ],[ ],0

+c 1 [ ] [ ],[ ], ,0 0 0 [ ]

+1 2 x

x+


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New solutions in terms of hypergeometric functions for 2nd order nonlinear

equations


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Second order nonlinear equations appear in the mathematical formulation of problems in many areas.

The methods of symmetries and integrating factors, implemented in Maple in previous releases, are the

most important ones. However, large ODE classes escape these methods or are more properly solved

by other means. For example the class of 2nd ordernonlinearODEs that can be obtained by reducingthird orderlinearODEs exploiting their scale invariance, admits explicit and tidy solutions that can

only be obtained if that relationship - instead of the symmetries - is unveiled. For Maple 12, a related

new algorithm was developed and implemented for solving that class, defined around the nonlinearequation


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> diff(y(x),x,x) = -(3*y(x) + c[2](x))*diff(y(x),x) - y(x)^3 - c[2](x)*y(x)^2 - c[1](x)*y(x) - c[0](x);


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=y'' ( )+3y ( )c2

x y' y3 ( )c2

x y2 ( )c1

x y ( )c0

x


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where the c[j](x) are arbitrary functions. The solution to this equation can be expressed in terms of the

solutions of a third order linear equation:


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> dsolve(%);


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=y( )DESol ,{ }+ + +( )c

0x ( )_Y x ( )c

1x _Y' ( )c

2x _Y'' _Y''' { }( )_Y x

x

( )DESol ,{ }+ + +( )c0

x ( )_Y x ( )c1

x _Y' ( )c2

x _Y'' _Y''' { }( )_Y x


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Note the derivative of a DESol structure in the numerator of the right-hand side. This nonlinear ODE

that the system can now solve generates an ODE class when it is transformed using


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> x -> F(x), y -> P(x)*y;


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,x ( )F x y ( )P x y


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The actual family of equations that is now solvable in Maple 12, for any value of the functions F(x) and

P(x), is


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> PDEtools:-declare((F, P)(x));


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( )F x will now be displayed as F


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( )P x will now be displayed as P


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> diff(y(x), x,x) = ((diff(F(x), x,x)*P(x) -2*diff(F(x),x)*diff(P(x),x) - diff(F(x),x)^2*c[2](F(x))*P(x))/diff(F(x),x)/P(x) -3*diff(F(x),x)*P(x)*y(x))*diff(y(x),x) - diff(F(x),x)^2*P(x)^2*y(x)^3- diff(F(x),x)*(3*diff(P(x),x) + c[2](F(x))*P(x)*diff(F(x),x))*y(x)^2+ (diff(F(x), x,x)*diff(P(x),x) - diff(F(x),x)*diff(P(x), x,x) -

diff(F(x),x)^2*c[2](F(x))*diff(P(x),x) - c[1](F(x))*P(x)*diff(F(x),x)^3)/diff(F(x),x)/P(x)*y(x) - c[0](F(x))*diff(F(x),x)^2/P(x);


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y''

F'' P 2F' P' F' 2 ( )c2

F P

F' P3F' P y y' F' 2P2y3 F'( )+3P' ( )c

2F P F' y2 =

( ) F'' P' F' P'' F' 2 ( )c2

F P' ( )c1

F P F' 3 y

F' P

( )c0

F F' 2

P+


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The new algorithm recognizes whether a given equation is of this form and if so it computes the values

of F(x), P(x) and the c[j](x), and with them constructs a 3rd orderlinearODE equivalent to the given

2nd ordernonlinearequation. This algorithm is automatically combined by dsolve with the new

algorithms for computing hypergeometric solutions for 3rd order linear ODEs, resulting in newsolutions available in Maple 12 for a large number of 2nd order nonlinear ODEs that were formerly

considered "unsolvable".


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Examples


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This second order nonlinear ODE is solved by transforming it into a third order linear ODE that admits

Liouvillian (trigonometric) solutions


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> diff(y(x), x,x) = (6/(x^2+1)^2*x*y(x) - (3*x^2-1)/(x^2+1)/x)*diff(y(x),x) - 4*x^2/(x^2+1)^4*y(x)^3+4*x^2/(x^2+1)^4;


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=y'' +

6x y

( )+x2 12

3x2 1( )+x2 1 x

y'4x2y3

( )+x2 14

4x2

( )+x2 14


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> dsolve(%);


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y 2_C1 e

1

+x2 1_C2 e

1

2 ( )+x2 1 cos

3

2 ( )+x2 1

=

_C2 e

1

2 ( )+x2 1

sin3

2 ( )+x2 13 e

1

2 ( )+x2 1

sin3

2 ( )+x2 1

e

1

2 ( )+x2 1

cos

3

2 ( )+x2 13+ 2 _C1 e

1

+x2 1

2_C2 e

1

2 ( )+x2 1 cos

3

2 ( )+x2 12 e

1

2 ( )+x2 1 sin

3

2 ( )+x2 1+ +


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The following nonlinear equation is solved the same way, but the related third order linear ODE only

admits hypergeometric solutions and so is solved using the corresponding new algorithms for that

problem in Maple 12


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> diff(y(x), x,x) = -3*diff(y(x),x)*y(x)-y(x)^3+1/x;


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=y'' +3 y' y y31

x


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> dsolve(%);


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y1

12

+24 x

2

hypergeom , ,[ ]

,

3

22

x2

8x4

hypergeom , ,[ ]

,

5

23

x2

8_C1

=

+12 x

hypergeom , ,[ ]

,

1

2

3

2

x2

84 x3

hypergeom , ,[ ]

,

3

2

5

2

x2

8_C2+

24

MeijerG , ,[ ],[ ] [ ]

,

, ,2

1

20 [ ]

x2

8

24

MeijerG , ,[ ],[ ] [ ]

,

, ,1

1

20 [ ]

x2

8+ x

_C1 x2

hypergeom , ,[ ]

,

3

22

x2

8_C2 x

hypergeom , ,[ ]

,

1

2

3

2

x2

8+

MeijerG , ,[ ],[ ] [ ]

,

, ,1

1

2 0 [ ] x2

8+


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New solutions in terms of hypergeometric functions for the 1st order AIL Abel

class of equations


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A new algorithm, solving the entire Abel Inverse Linear (AIL) class of equations (see references) in

terms of hypergeometric functions, was developed and implemented for Maple 12. This class of

equations is the one you obtain from the standard AIL equation


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> diff(y(x),x) = (a[0] + a[1]*y(x) + a[2]*y(x)^2 + a[3]*y(x)^3)/((s[0] + s[1]*x)*y(x) + r[0]+r[1]*x);


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=y'+ + +a

0a

1y a

2y2 a

3y3

+ +( )+s0

s1

x y r0

r1

x


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under general transformations


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> x -> F(x), y -> (P(x)*y + Q(x)*y)/(S(x)*y + T(x)*y);


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,x ( )F x y+( )P x y ( )Q x y+( )S x y ( )T x y


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where F is rational and R, P, Q, S, T are arbitrary. The novelty with regards to previous Maple releases

where this same class was partially solved is that in Maple 12 any equation obtained using a rational

form ofF(x) is now solved.


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Examples


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The following equation, involving only rational coefficients, admits a solution that involves only

combinations of fractional and integer powers after all hypergeometric functions got simplified


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> diff(y(x),x) = 1/2*1/x*(-x+5/2)/(-x+2)^3*y(x)^3+(1/2*x-3)/x/(2-x)^2*y(x)^2+1/x/(-x+2)*y(x);


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=y' + +

+x

5

2y3

2 x ( ) +x 2 3

x

23 y2

x ( ) +x 2 2y

x ( ) +x 2


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> dsolve(%);


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_C1 625 10 6 + 2x 4 y

+8 ( )+4 y x

+ +

+

2

5y x3

+ y

2 8

5

22

5y x2

+

24

5y

49

10y2

8

5x 6y2 54

+ ( ) +2 2y x 4 5y

+8 ( )+4 y x

( )/5 2

x ( ) +8 ( )+4 y x 2 0=


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> odetest(%, %%); # verify this result - see ?odetest


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0


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Partial Differential Equations (PDEs)


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Six new commands are available in PDEtools: Laplace, Euler, ConservedCurrents,

ConservedCurrentTest, IntegratingFactors, and IntegratingFactorTest.


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The Laplace command explores Laplace invariants to compute the general solution in terms of

arbitrary functions to a single 2nd order linear PDE in two independent variables.


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Examples


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> PDEtools:-undeclare(prime);


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There is no more prime differentiation variable; all derivatives will be displayed as \

indexed functions


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> PDEtools:-declare(u(x,y), _F1(x), _F2(y));


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( )u ,x y will now be displayed as u


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( )_F2 y will now be displayed as _F2


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>pde := diff(u(x, y), x, y)-2*u(x, y)/(x+y)^2;


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:=pde u,x y

2 u

( )+x y 2


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> sol := PDEtools:-Laplace(pde, u(x,y));


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:=sol =u1

2

+ ( )+x y _F1x

( )+x y _F2y

2_F2 2_F1

+x y


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This new Laplace command is automatically invoked bypdsolve, so the same solution above, in terms

of two arbitrary functions of one variable,_F1(x) and_F2(y), is returned bypdsolve. PDE solutions

can be verified usingpdetest


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>pdetest(sol, pde);


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0


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Given a system = 0 consisting ofNequationspde[n], n = 1..N, involving Mindependent variablesx[1],x[2], ... =X, and where the dependent variables are u[1], u[2], ... = U, with dUdenoting the set

of partial derivatives ofU, conserved currents are expressionsJ[ , m](X, U, dU) each of whichsatisfy: d/dx[1]J[ , 1](X, U, dU) + d/dx[2]J[ , 2](X, U, dU) + ... = DivergentJ[ , m] = 0,where m = 1..Mand d/dx[m] represents the total derivative with respect tox[m]. The conserved

currents coincide with the traditional first integrals when there is only one independent variable, so

that is a system of ODEs.


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The pair of commands ConservedCurrents and ConservedCurrentTest respectively compute these

conserved currents and verify whether a given expression is or is not a conserved current of a given

PDE system.


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Analogously, thegeneralizedintegrating factors are expressions [ , n](X, U, dU) such that [ , n]pde[n] = DivergenceJ[ ] = 0, soJ[ ] is a conserved current. The pair of commandsIntegratingFactors and IntegratingFactorTest can respectively compute these generalized

integrating factors and verify the result for correctness.


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Examples


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Consider the following PDE "system" consisting of a single pde


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> with(PDEtools):


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U := diff_table(u(x,t)):


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declare(U[]);


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( )u ,x t will now be displayed as u


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>pde := U[t,t] + U[x,x] + U[x]*U[];


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:=pde + +u,t t

u,x x

ux

u


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The conserved currents ofpde are computed as follows and depend on arbitrary functions


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> J[alpha] := ConservedCurrents(pde);


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J =( )_J1 , , , ,x t u ux ut + + +ux ( ) _F3 , ,x t u ut d

u

+_F3t

_a _a ( ) _F5 ,x t ,

:=

=( )_J2

, , , ,x t u ux

ut

+ + + +( ) _F3 , ,x t u ux

ut

d

u

_F3x

_a d

_F5x t ( )_F6 x

,

=( )_J1

, , , ,x t u ux

ut

+ + +t ux

( ) _F3 , ,x t u ut

d

u

+_F3t

_a t _a ( ) _F5 ,x t ,

=( )_J2

, , , ,x t u ux

ut

+ + + +( ) _F3 , ,x t u ux

t ut

d

u

_F3x

_a u d_F5x t ( )_F6 x


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To verify these results use


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>map(ConservedCurrentTest, [J[alpha]], pde);


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[ ],{ }0 { }0


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The conserved currents are related to the generalized integrating factors [ ] via [ , n]pde[n] = DivergenceJ[ ] = 0. These are the [ ] corresponding to theJ[ ] computed above


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>mu[alpha] := IntegratingFactors(pde);


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:= ,[ ]=( )_ 1 , , , ,x t u ux ut t [ ]=( )_ 1 , , , ,x t u ux ut 1


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To verify for correctness these integrating factors use


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>map(IntegratingFactorTest, [mu[alpha]], pde);


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[ ],{ }0 { }0


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Numerical Differential Equations


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There were a significant number of improvements made for numerical ODE solution for this release.


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Events


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The default ODE and DAE IVP integrators now have the ability to handle user-defined events. These

events are an extension to stop conditions (which are now deprecated), and have the ability to execute

code when an event occurs, and the ability to perform event iteration (i.e. when one event triggers

another).


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In addition, events have several interactive features, such as the ability to query the event that fired, and

obtain the solution value at the time the event occurred.


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For more information, see dsolve,numeric,Events.


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Parameters


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The ODE and DAE IVP integrators have been extended to handle parametrized problems. This means

that a dsolve procedure can be formed for a class of ODE or DAE, and the parameters can be adjusted,

and different solutions obtained interactively.


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For more information, see dsolve,numeric,interactive.


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Discrete Variables


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The default ODE and DAE IVP integrators now allow the definition of discrete variables as part of the

problem description. These discrete variables can be of float, integer or boolean type, and are most

useful in combination with event handling.


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For more information, see dsolve,numeric,Events.


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See Also


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dsolve,numeric, dsolve,numeric,Events, dsolve,numeric,interactive, Index of New Maple 12 Features,

PDEtools


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>


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>

Documents

Updates to Differential Equation (de) Solvers In